Classical and Bayesian estimation for type-I extended-F family with an actuarial application

In this work, a new flexible class, called the type-I extended-F family, is proposed. A special sub-model of the proposed class, called type-I extended-Weibull (TIEx-W) distribution, is explored in detail. Basic properties of the TIEx-W distribution are provided. The parameters of the TIEx-W distribution are obtained by eight classical methods of estimation. The performance of these estimators is explored using Monte Carlo simulation results for small and large samples. Besides, the Bayesian estimation of the model parameters under different loss functions for the real data set is also provided. The importance and flexibility of the TIEx-W model are illustrated by analyzing an insurance data. The real-life insurance data illustrates that the TIEx-W distribution provides better fit as compared to competing models such as Lindley–Weibull, exponentiated Weibull, Kumaraswamy–Weibull, α logarithmic transformed Weibull, and beta Weibull distributions, among others.

5. Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article's retracted status in the References list and also include a citation and full reference for the retraction notice.
Answer: Thanks for this comment. We have removed two references as suggested by Reviewer #1. Some references has been added in the application section (Section 7) because we have added some new models in the application to compare them with the TIEx-W model as suggested by Reviewer #2. All changes in the revised version are highlighted in magenta color.

Reviewers' comments:
Reviewer #1: 1. The introduction must be separate from the methodology.
Answer: Thanks for this suggestion. We have improved accordingly. We have added a new section entitled ''The TIEx-F Family''.
2. The conclusion must be well written and organized.
Answer: Thanks for this suggestion. The conclusion section is corrected accordingly.
3. Some tables do not cite; they must be cited ("The parameters estimate …. Table??. The goodness… in Table??.") 4. The authors should provide the other basic functions of the TIEx-W distribution, including the survival and hazard rate functions, along with plot(s) showing some possible shapes of the hazard rate function for different settings of the model's parameters. Also present some discussions on the plots and whether the hazard rate function is relevant in modelling insurance data.
Answer: Thanks for this suggestion. We have provided the required discussions, functions, figures, and discussions. Please see Equations (8)  Answer: Thanks for your comment. Yes, the quantile function has a closed form expression, we provided it in Equation (10)  10. In section 7, discuss the reasons/justifications for choosing gamma priors for the three parameters of TIEx-W model.
Answer: Thanks for this comment. It is well-known that the gamma priors provide flexible approach to handle estimation procedure in both informative and non-informative.
11. The authors need to provide some discussions on the Bayesian results.
Answer: Thanks for this comment. We have provided the required discussion.
12. I also expect to see some comparisons between the results from the classical estimation methods and the Bayesian results either by plotting the density curves, CDF curves for the trained TIEx-W model from all the estimation methods employed or the p-p plots. This is essential to identify the best estimation method(s) in solving real-world problems.
Answer: Thanks for this comment. We have provided a new table which includes the estimates of the TIEx-W parameters using all estimation methods.
We have also provided the density plots and PP plots for all methods.
13. How the estimated values of the parameters can be translated in the real-life situations?
Answer: Thanks for this comment. Our aim is to show the flexibility of the proposed TIEx-W distribution in modelling insurance data. Based on the fitted TIEx-W model, we can answer some questions such as the probability of having more than claims per day.

Reviewer #3:
The authors have put in considerable effort in explaining their research aim and finding but need to add some comments on the tables and charts rather than leaving them blank. The functional for of the estimators should also be provided even if the procedure can be verified by readers.
Answer: Thanks for this comment. We have provided the required comments about tables and figures. The functions for all proposed estimators are provided in the paper. We have also added some new results as suggested by the two other reviewers.